Let's break down the problem step by step:
### Step 1: Calculate the volume of the Sun
The formula to calculate the volume [tex]\( V \)[/tex] of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Given:
- The radius of the Sun [tex]\( r_{\text{Sun}} = 695,000 \text{ km} \)[/tex]
Plugging in the values:
[tex]\[ V_{\text{Sun}} = \frac{4}{3} \pi (695,000)^3 \][/tex]
The volume of the Sun in scientific notation with 3 decimal places in the mantissa is:
[tex]\[ V_{\text{Sun}} = 1.406 \times 10^{18} \text{ km}^3 \][/tex]
### Step 2: Calculate the radius of the Earth
Given:
- The volume of the Earth [tex]\( V_{\text{Earth}} = 1.1 \times 10^{12} \text{ km}^3 \)[/tex]
The formula for the volume of a sphere is rearranged to solve for the radius [tex]\( r \)[/tex]:
[tex]\[ r = \left( \frac{3V}{4 \pi} \right)^{\frac{1}{3}} \][/tex]
Plugging in the values:
[tex]\[ r_{\text{Earth}} = \left( \frac{3 \times 1.1 \times 10^{12}}{4 \pi} \right)^{\frac{1}{3}} \][/tex]
The radius of the Earth is:
[tex]\[ r_{\text{Earth}} \approx 6403.8 \text{ km} \][/tex]
### Step 3: Calculate how many times the Sun's radius is larger than Earth's radius
[tex]\[ \text{Sun to Earth radius ratio} = \frac{r_{\text{Sun}}}{r_{\text{Earth}}} = \frac{695,000}{6403.8} \][/tex]
The Sun's radius is approximately:
[tex]\[ \text{Sun to Earth radius ratio} \approx 108.5 \][/tex]
### Step 4: Calculate how many times the Sun's volume is larger than Earth's volume
[tex]\[ \text{Sun to Earth volume ratio} = \frac{V_{\text{Sun}}}{V_{\text{Earth}}} = \frac{1.406 \times 10^{18}}{1.1 \times 10^{12}} \][/tex]
The Sun's volume is approximately:
[tex]\[ \text{Sun to Earth volume ratio} \approx 1,278,351.7 \][/tex]
### Final Answer
1. The volume of the Sun is:
[tex]\[ 1.406 \times 10^{18} \text{ km}^3 \][/tex]
2. The radius of the Earth is:
[tex]\[ 6403.8 \text{ km} \][/tex]
3. The Sun's radius is:
[tex]\[ 108.5 \text{ times the Earth's radius} \][/tex]
4. The Sun's volume is:
[tex]\[ 1,278,351.7 \text{ times the Earth's volume} \][/tex]
